reserve k,n,m,l,p for Nat;
reserve n0,m0 for non zero Nat;
reserve f for FinSequence;
reserve x,X,Y for set;
reserve f1,f2,f3 for FinSequence of REAL;
reserve n1,n2,m1,m2 for Nat;
reserve I,j for set;
reserve f,g for Function of I, NAT;
reserve J,K for finite Subset of I;

theorem Th27:
 for j being object holds J = {j} implies Sum(f|J) = f.j
proof let j be object;
  consider f9 be FinSequence of REAL such that
A1: Sum(f|J) = Sum f9 and
A2: f9 = (f|J)*canFS(support(f|J)) by UPROOTS:def 3;
  assume
A3: J = {j};
  then
A4: j in J by TARSKI:def 1;
  then j in I;
  then j in dom f by FUNCT_2:def 1;
  then J c= dom f by A3,ZFMISC_1:31;
  then
A5: dom(f|J) = J by RELAT_1:62;
  per cases by A3,ZFMISC_1:33;
  suppose
A6: support(f|J) = {};
    now
      assume f.j <> 0;
      then (f|J).j <> 0 by A4,FUNCT_1:49;
      hence contradiction by A6,PRE_POLY:def 7;
    end;
    hence Sum(f|J) = f.j by A1,A2,A6,RVSUM_1:72;
  end;
  suppose
    support(f|J) = J;
    then canFS(support(f|J)) = <*j*> by A3,FINSEQ_1:94;
    then f9 = <*(f|J).j*> by A4,A5,A2,FINSEQ_2:34;
    then f9 = <* f.j *> by A4,FUNCT_1:49;
    hence Sum(f|J) = f.j by A1,RVSUM_1:73;
  end;
end;
